Keepers arrived to find the new arrival nestling in the arms of its mother.
Photographs taken just hours after the birth on Wednesday show Kala - a nine-year-old western lowland gorilla - cradling the newborn. Staff said both were "doing well".
The zoo said Kala gave birth naturally with the baby's father, Jock, nearby. Her first baby died last year a week after it was born.
Lynsey Bugg, the zoo's curator of mammals, said: "We knew we were having a baby gorilla due and we've been on baby watch for a little while.
"On Tuesday Kala looked nice and comfortable and not causing us any concerns or worries.
"I came in [on Wednesday] morning to find a brand new baby in the house. It was lovely."
She said staff had been "on tenterhooks" following the death of Kala's first baby last September, a week after she underwent an emergency caesarean.
"It is so lovely that she was able to give birth naturally and baby and mum are really well.
"She's a very attentive mother and very nurturing and you see lots of suckling from the baby, and the baby looks really strong and a good size."
It will be a while before the zoo knows if the baby is male or female, Ms Bugg said.
"They are not all that easy to sex and we want to have a few looks before we are certain."
The new gorilla joins a troop of six at the zoo, which are part of a breeding programme.
The western lowland gorilla is classed as critically endangered, with some estimates putting the number left in the wild at about 100,000.
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